The precision of typical thermometers consisting of $N$ particles is shot
noise limited, improving as $\sim1/\sqrt{N}$. For high precision thermometry
and thermometric standards this presents an important theoretical noise floor.
Here it is demonstrated that thermometry may be mapped onto the problem of
phase estimation, and using techniques from optimal phase estimation, it
follows that the scaling of the precision of a thermometer may in principle be
improved to $\sim1/N$, representing a Heisenberg limit to thermometry.Comment: 4 page

Stace, Thomas M.

English

arXiv

Thomas M. Stace

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Physical Review A

Quantum limits of thermometry

The precision of typical thermometers consisting of N particles scales as ~1/√N. For high-precision thermometry and thermometric standards, this presents an important theoretical noise floor. Here it is demonstrated that thermometry may be mapped onto the problem of phase estimation, and using techniques from optimal phase estimation, it follows that the scaling of the precision of a thermometer may in principle be improved to ~1/N, representing a Heisenberg limit to thermometry. © 2010 The American Physical Society

University of Queensland eSpace

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PHYSICAL REVIEW A 82, 011611(R) (2010)
Quantum limits of thermometry
Thomas M. Stace*
Department of Physics, University of Queensland, Brisbane, Queensland 4072, Australia
(Received 18 May 2010; published 30 July 2010)
The precision of typical thermometers consisting of N particles scales as ∼1/√N . For high-precision
thermometry and thermometric standards, this presents an important theoretical noise floor. Here it is demonstrated
that thermometry may be mapped onto the problem of phase estimation, and using techniques from optimal phase
estimation, it follows that the scaling of the precision of a thermometer may in principle be improved to ∼1/N ,
representing a Heisenberg limit to thermometry.
DOI: 10.1103/PhysRevA.82.011611 PACS number(s): 03.75.Dg, 07.20.Dt, 03.67.Ac
Temperature is a surprisingly subtle concept: it is easily
perceived, yet it took 300 years to formalize the concept in
operationally useful terms [1]. One of the principal challenges
in understanding temperature and thermodynamics was the
development of practical and accurate thermometers capa-
ble of measuring thermodynamic temperature. Indeed, this
remains an outstanding issue in modern thermometry: the
relative uncertainty in the CODATA value for Boltzmann’s
constant, kB , is ur = 1.7 × 10−6 [2]. This makes it one of
the least precisely measured fundamental constants, surpassed
only by Newton’s gravitational constant for imprecision.
With a few exceptions, thermometers generally consist
of a system of low thermal mass which is brought into
thermal contact with the system of interest, the bath. Following
thermalization, some state parameter of the thermometer is
measured from which the temperature of the thermometer,
and thus the bath, is inferred. A low thermal mass is required
to minimize the heat transferred between the bath and the
thermometer compared to the total internal energy of the bath,
so that the temperature of the bath is only slightly perturbed.
For thermometric standards, the requirement of a small
thermal mass is less important. The current CODATA value
of kB is derived from measurements of a device embedded
in an ice-water bath held at the triple point of water (TPW),
which is defined to be exactly TTPW = 273.16 K [3]. At the
triple point, the temperature of the bath is independent of
the internal energy of the bath, so the relative sizes of the
thermometer and the bath are largely irrelevant. In practice,
however, having a thermometer of a small spatial extent is
important for technical reasons: minimizing temperature and
pressure variations is easier over a smaller volume.
Given the inherent advantages of small thermometers for
both thermometry and thermometric standards, it is natural to
enquire about the ultimate constraints that thermometer size
imposes on the precision in measuring β−1 = kBT . In what
follows, I first discuss the intrinsic noise limit of thermalizing
thermometers by reviewing the fact that the precision of such
a thermometer comprising N particles cannot scale better
than σβ ∼ 1/
√
N , assuming only that the total internal energy
is an extensive parameter. By analogy with the noise floor
in optical systems, this scaling is termed the “shot-noise”
scaling for the remainder of this paper. This shot-noise scaling
*stace@physics.uq.edu.au
is demonstrated in a very simple model consisting of N
independent atoms.
The central result of this paper is that shot-noise scaling
is not a fundamental limit to thermometry. By means of an
explicit construction, I show that Heisenberg-limited scaling
is achievable, that is, σβ ∼ 1/N . This represents a new class
of thermometer that does not attain thermal equilibrium with
the bath it is measuring. Instead, the temperature of the bath
is mapped onto a phase factor in the quantum state of the
thermometer, which is then estimated using interferometric
methods. Exploiting techniques developed for optimal phase
estimation, it follows that this nonthermalizing thermometer is
capable of attaining Heisenberg-limited scaling. The construc-
tion is built on an extension of the model of N independent
atoms and serves principally to demonstrated the existence of
Heisenberg-limited thermometry.
To derive the shot-noise limit in thermalizing thermometers,
consider the thermal state of a thermometer with internal
Hamiltonian Hth
ρ = e−βHth/Z, (1)
where Z = Tr{e−βHth}. Assuming that the average internal
energy of the thermometer is extensive, it is given by ¯E =
− ∂ ln Z
∂β
= Nε¯(β), where ε¯ is the average internal energy
per particle, which is independent of N . Since ε¯(β) is a
monotonically increasing function of temperature, the sample
standard deviation of ε¯ and β are related by the identity
σβ = σε/ε¯′, (2)
where ε¯′ = |dε¯/dβ|. To calculate σε, note that the sample
variance in the total internal energy is given by
σ 2E =
∂2 ln Z
∂β2
= −∂
¯E
∂β
= Nε¯′, (3)
which demonstrates that the variance in the total internal en-
ergy of the thermometer is extensive. The relative uncertainty
in the energy per particle is thus
σε
ε¯
= σE
¯E
= 1√
N
√
ε¯′
ε¯
. (4)
Equations (2) and (4) together imply that
σβ = 1√
N
1√
ε¯′
, (5)
1050-2947/2010/82(1)/011611(4) 011611-1 ©2010 The American Physical Society
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THOMAS M. STACE PHYSICAL REVIEW A 82, 011611(R) (2010)
thus establishing the shot-noise limit on extensive, thermaliz-
ing thermometers.
This result may also be derived without explicit reference
to any measurement parameter by computing the Fisher
information, F (β), for a thermal state. Since the thermal state
is diagonal in the energy eigenbasis, it is straightforward to
show that F (β) = σ 2E . It is then apparent that Eq. (5) saturates
the Fisher inequality σβ  1/F (β) [4,5], and so Eq. (5) indeed
represents the ultimate limit of the precision of a thermalizing
thermometer.
This scaling is potentially a significant issue for high-
precision thermometry. One promising avenue currently being
pursued to improve the precision of measurements of kB
relies on Doppler gas thermometry, in which high-precision
spectroscopy of a gas in thermal equilibrium with a TPW
bath reveals a Maxwell-Boltzmann distribution of velocities
[6–8], whose width is a direct measure of kBTTPW. In recent
Doppler thermometry experiments in alkali vapors, the atomic
flux through a beam ∼10 cm long and 2 mm in diameter is
˙N ∼ 1015 atoms/s [9]. The limit to the precision of such a
thermometer is then σβ ∼ ( ˙Nτ )−1/2 ≈ 10−7.5τ 1/2, where τ is
the integration time. At 1 s this sets a maximum precision
in measurements of kB of 1 part in 107.5, which is about 1.5
orders of magnitude better than the current CODATA estimates
for kB .
The shot-noise scaling of a thermalizing thermometer is
exhibited by a toy model of N identical, noninteracting, two-
level atoms, each with energy eigenstates |g〉 and |〉. The
Hamiltonian is simply
Hth =
N∑
j=1
Hj, where Hj = |〉j 〈|. (6)
To simplify the analysis, suppose that the thermometer atoms
first come into thermal equilibrium with the bath, then they
are adiabatically isolated from the bath, and finally, the total
energy of the atoms is measured to infer the temperature. The
partition function for the thermometer atoms is
Z = Tr {e−βHth} = (1 + e−β)N, (7)
and the internal energy and its variance are
¯E = 〈Hth〉 = −∂ ln Z
∂β
= N 
1 + eβ , (8)
σ 2E =
〈
H 2th
〉− 〈Hth〉2 = ∂
2 ln Z
∂β2
= N 
2eβ
(1 + eβ)2 , (9)
which are explicitly extensive. By inspection, ε¯ = 1+eβ , and
this, along with the relative uncertainty in β, are plotted in
Fig. 1.
This scaling behavior, σβ ∝ 1/
√
N , is strongly reminiscent
of the standard quantum limit (SQL) to phase estimation of
an unknown phase, φ, in one branch of an interferometer.
The SQL is a consequence of shot-noise intrinsic to the states
incident on the input ports of the interferometer. In particular,
if one input port of the interferometer is illuminated with a
Fock state |n〉 of n photons, or with a coherent pulse |α〉 with
an expected number of photons |α|2 = n, φ may be estimated
to a precision that scales as ∼1/√n.
0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
0
2
4
6
8
10
β ∋
ε
∋
N
σΒ
Β
FIG. 1. (Color online) Energy per particle ε¯ and relative sample
uncertainty in β, scaled by
√
N , given by Eq. (5), for a toy
thermometer consisting of N independent, two-level systems.
It is possible to beat the SQL scaling for phase estimation
by generating either squeezed states or N00N states in the
interferometer [5,10]. Indeed, these nonclassical states saturate
the ultimate Heisenberg limit for phase estimation, with a
precision that scales as 1/n.
Given this, it is natural to ask whether one can construct a
“Heisenberg-limited thermometer,” whose precision improves
as 1/N , just as the precision of Heisenberg-limited phase
estimation improves as 1/n. In what follows, this question
is answered in the affirmative.
From the preceding analysis it follows that any extensive,
thermalizing thermometer will be subject to SQL-like scaling.
To beat this scaling it is therefore necessary to relax either
extensivity (of the internal energy) or thermalization (of the
thermometer). The former seems difficult, since it requires
interactions within the thermometer to dominate at all scales.
In contrast, the construction presented here demonstrates
that it is possible to do thermometry without thermalizing
the thermometer1 and that such thermometers may indeed
attain the Heisenberg limit 1/N . This is accomplished by
mapping the problem of measuring the temperature of a bath
to the problem of estimating an unknown phase, then using
techniques for optimal phase estimation.
To demonstrate the function of a nonthermalizing ther-
mometer, a toy model for the thermometer, the bath, and their
interaction is now presented. The thermometer is as already
described: a collection of N noninteracting, two-level atoms.
The bath is similar, consisting of M  N such atoms. The
form of interaction between the thermometer and the bath is
crucial:
Hint = α
N∑
j=1
M∑
k=1
|〉j 〈| ⊗ |〉k〈|, (10)
where index j refers to atoms in the thermometer and k refers
to atoms in the bath. The thermometer atoms are assumed to be
bosonic, and creation and annihilation operators a†i and ai are
introduced for atoms in mode i acting on the atomic vacuum
state |0〉.
1Existing pyrometers measure temperatures that far exceed the
melting point of the device components.
011611-2
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QUANTUM LIMITS OF THERMOMETRY PHYSICAL REVIEW A 82, 011611(R) (2010)
Bath
B
1
4
3
6
5
2
Atomic
source
FIG. 2. An atomic interferometer with the bath in one branch.
Atoms from the thermometer are input into mode 1 and detected in
modes 5 and 6. Mode 2 is the vacuum port.
The bath is assumed to be in a thermal state at some
unknown temperature, ρB = e−βHbath/Zbath, isolated from any
environment, so that its internal energy is constant during the
thermometry, and embedded in an interferometer, as depicted
in Fig. 2. Suppose that a single thermometer atom is prepared
in state |〉 and introduced to the input port, 1, of the atomic
beam splitter, which effects the map
a
†
1 → (a†3 + ia†4)/
√
2. (11)
As the wave function propagates through the interferometer,
the component in the branch containing the bath acquires a
phase φB = αmτ , where m is the the number of excited atoms
in the bath and τ is the interaction time.
Upon repeating this N times, φB and m = φB/(ατ ) can be
inferred from the output statistics. This yields an estimate of
both φB and m with a precision given by
σ (SN)m =
σ
(SN)
φB
ατ
= 1
ατ
1√
N
. (12)
Since 〈m〉 = M/(1 + eβ), the precision in the measured
temperature is shot noise limited:
σ
(SN)
β = σ (SN)m
/|d〈m〉/dβ| = 1 + e
β
eβ
1
ατ
1√
N
. (13)
While this scaling is no better than that of a thermalizing
thermometer, this construction demonstrates that measuring
the bath temperature using an N -atom thermometer can be
mapped onto the task of phase estimation using N atoms. It
follows that techniques from optimal phase estimation lead to
improved precision in thermometry.
In particular, suppose that the linear atomic beam splitters
shown in Fig. 2 are replaced with nonlinear beam splitters that
effect the N -atom map
(a†1)N |0〉 → [(a†3)N + i(a†4)N ]|0〉, (14)
which maps an N photon fock state onto a N00N state in
which all N atoms take the same branch. As the atoms in the
upper branch interact with the bath atoms, according to the toy
Hamiltonian already given, a phase NφB is accumulated on
mode 4, leaving the atom interferometer in the state
[(a†3)N + ieiNφB (a†4)N ]|0〉. (15)
As shown in [11], this state has maximal Fisher information
with respect to φB , so φB can be measured at the Heisenberg
limit, that is, with precision σ (H )φB = 1/N . This is accomplished
by measuring the observable
ˆAN = (a†3)N |0〉〈0|(a3)N + (a†4)N |0〉〈0|(a4)N. (16)
It follows that the temperature can be measured with a
precision that scales as 1/N :
σ
(H )
β = σ (H )m
/|d〈m〉/dβ| = 1 + e
β
eβ
1
ατ
1
N
. (17)
This demonstrates an N -atom thermometer that can indeed
attain the Heisenberg limit, thereby beating the shot-noise
limit. Implementing the specific toy model presented here
poses technical challenges, not least the creation of N00N
states. Nevertheless, its primary purpose in this paper is to
demonstrate theoretically the existence of Heisenberg-limited
thermometry.
In this construction, there were two crucial assumptions.
First, the bath was assumed to be isolated, so that both m and
M remain fixed during the measurement. If m varies during the
measurement, then fluctuations in NφB will lead to dephasing
of the N00N state, and the precision will be reduced. If M
varies, then clearly m will also, but so will the estimate of the
total internal energy of the system.
Second, the interaction between the atoms in the bath and
those in the thermometer must be precisely engineered, to be
of the form given in Eq. (10), which is diagonal in the energy
eigenbasis. This means that the thermometer is not universal,
since it must be designed to be compatible with the bath.
This is in contrast to thermalizing thermometers, which attain
thermal equilibrium for a very large class of bath systems.
Indeed, it suggests that for thermometric standards, in which
the bath is chosen for convenient properties of its triple point,
it is a problem of extreme complexity to engineer a coupling
of the form required. While there may be alternative models
of thermometry that also attain the Heisenberg limit, it is not
clear that these two assumptions can be relaxed.
As discussed previously, the temperature of an isolated
thermal bath is intrinsically defined only to a precision δβB ∼
1/
√
M . It follows that the maximum precision required of a
thermometer is limited by the size of the bath it is measuring:
no advantage is gained by using a thermometer capable of
a higher precision or by extending the integration time for
improved averaging. The fact that thermalizing thermometers
are shot noise limited, δ(SN)βT ∼ 1/
√
N , implies that to reach the
intrinsic precision determined by δβB , such thermometers need
to be at least as large as the bath they are measuring; that is,
δ
(SN)
βT
= δβB ⇒ N ∼ M. (18)
This condition competes with the requirement that the ther-
mometer be much smaller than the bath, highlighting the
implicit tension between high-precision thermometry and
thermometer-induced perturbations of the bath.
In contrast, for a Heisenberg-limited thermometer, for
which δ(H )βT ∼ 1/N , the condition
δ
(H )
βT
= δβB ⇒ N2 ∼ M (19)
may be satisfied for N  M , and so the thermometer may be
much smaller than the bath.
011611-3
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Small-scale demonstrations of Heisenberg-limited ther-
mometry using the ideas discussed here may be realized in
atom or ion traps, for example, in a trap consisting of a
relatively few bath atoms (say, M <∼ 100) and even fewer
thermometer atoms (N ≈ √M <∼ 10), possibly leading to new
techniques for measuring the extremely low temperatures
achieved in these systems.
The existence of Heisenberg-limited thermometers having
been established, by explicit construction, there are several
natural questions that follow from this work. It seems plausible
that the Heisenberg limit is the ultimate limit of thermometry,
however, the constructive arguments given here do not
establish this definitively. Further, the constraints on the bath
and the interaction Hamiltonian in the toy model required
to attain the Heisenberg limit of thermometry are poten-
tially difficult to engineer. There may be other implementa-
tions that relax these constraints, leading to more practical
Heisenberg-limited thermometers. Finally there is an intrigu-
ing connection with thermal clocks [12] that remains to be
explored.
This work was funded by the Australian Research Council.
I thank Gerard Milburn, Andre Luiten, Gar-Wing Truong,
Eric May, and Geoff Pryde for helpful conversations.
[1] H. Chang, Inventing Temperature: Measurement and Scientific
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2010).
[6] T. M. Stace and A. N. Luiten, Phys. Rev. A 81, 033848 (2010).
[7] C. Daussy, M. Guinet, A. Amy-Klein, K. Djerroud, Y. Hermier,
S. Briaudeau, C. J. Borde´, and C. Chardonnet, Phys. Rev. Lett.
98, 250801 (2007).
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D. Di Serafino, P. Laporta, and L. Gianfrani, Phys. Rev. Lett.
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011611-4


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